Catching a Drifting Tablet Press

A tablet press should make 500 mg tablets, but every tablet varies a little. Averaging small samples and plotting them against limits three standard errors from the target separates the everyday noise from a real change: a point outside the limits means stop and look.

How it works

X̄ control charts

In one sentence

With subgroups of size $n$ from a process with standard deviation $\sigma$ , the $\bar x$ chart for target $\mu_0$ has limits $\mu_0 \pm 3\sigma/\sqrt{n}$ . If the mean shifts to $\mu_0 + \delta\sigma$ , each sample signals with probability $1 - \Phi(3 - \delta\sqrt{n}) + \Phi(-3 - \delta\sqrt{n})$ , and the average run length to a signal is the reciprocal.

General formula

{\htmlData{sym=0}{\text{UCL}}} = {\htmlData{sym=2}{\mu_0}} + \frac{3{\htmlData{sym=3}{\sigma}}}{\sqrt{\htmlData{sym=4}{{n}}}}, \qquad {\htmlData{sym=1}{\text{LCL}}} = {\htmlData{sym=2}{\mu_0}} - \frac{3{\htmlData{sym=3}{\sigma}}}{\sqrt{\htmlData{sym=4}{{n}}}}, \qquad {\htmlData{sym=5}{\bar x_i}} \sim N\!\left({\htmlData{sym=6}{\mu}}, \frac{{\htmlData{sym=3}{\sigma}}^2}{\htmlData{sym=4}{{n}}}\right)

Noise and real change look alike point by point. Averaging shrinks the noise by √n, and limits at ±3 standard errors make false alarms rare, so a point outside them is worth investigating.

What the symbols mean

$\text{UCL}$
Upper control limit
Three standard errors above the target.
$\text{LCL}$
Lower control limit
Three standard errors below the target.
$\mu_0$
Target
The mean the process should run at: 500 mg here.
$\sigma$
Process standard deviation
How much single tablets vary: 6 mg here.
$n$
Sample size
Tablets averaged for each point.
$\bar x_i$
Sample mean
The average weight of sample i.
$\mu$
Process mean
Where the press is actually running, which can drift.

Each point is the mean $\bar x_i$ of a sample of $n$ tablets; by the central limit theorem it varies around the process mean $\mu$ with standard error $\sigma/\sqrt n$ .
While the process is on target ( $\mu = \mu_0$ ), $\bar x_i$ falls within $\mu_0 \pm 3\sigma/\sqrt n$ about 99.73% of the time, so a point outside is strong evidence something changed.
After sample 15 the lab shifts the mean by $\delta\sigma$ : how fast points start landing outside depends on the shift and on $n$ .
Larger samples shrink $\sigma/\sqrt n$ , pulling the limits in, so the same shift is caught sooner.

Learn the math

X̄ and R Control ChartsEng Stats · Lesson 27, Statistical Process Control